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Sequences of pyramidal numbers (1).



Abstract Shyam Sunder sun·der  
v. sun·dered, sun·der·ing, sun·ders

v.tr.
To break or wrench apart; sever. See Synonyms at separate.

v.intr.
To break into parts.

n.
A division or separation.
 Gupta Gupta (gp`tə), Indian dynasty, A.D. c.320–c.550, whose empire at its height encompassed much of N India. Ancient Indian culture reached a high point during this period.  [4] has defined Smarandache consecutive and reversed Smarandache sequences of Triangular numbers (Math.) the series of numbers formed by the successive sums of the terms of an arithmetical progression, of which the first term and the common difference are 1. See Figurate numbers, under Figurate.

See also: Triangular
. Delfim F.M.Torres and Viorica Teca [1] have further investigated these sequences and defined mirror and symmetric No difference in opposing modes. It typically refers to speed. For example, in symmetric operations, it takes the same time to compress and encrypt data as it does to decompress and decrypt it. Contrast with asymmetric.

(mathematics) symmetric - 1.
 Smarandache sequences of Triangular numbers making use of Maple system. One of the authors A.S.Muktibodh [2] working on the same lines has defined and investigated consecutive, reversed, mirror and symmetric Smarandache sequences of pentagonal numbers of dimension 2 using the Maple system. In this paper we have defined and investigated the s-consecutive, s-reversed, s-mirror and s-symmetric sequences of Pyramidal numbers (Math.) certain series of figurate numbers expressing the number of balls or points that may be arranged in the form of pyramids. Thus 1, 4, 10, 20, 35, etc., are triangular pyramidal numbers; and 1, 5, 14, 30, 55, etc., are square pyramidal numbers.

See also: Pyramidal
 (Triangular numbers of dimension 3.) using Maple 6.

[section] 1. Introduction

Figurate number A figurate number is a number that can be represented as a regular and discrete geometric pattern (e.g. dots). If the pattern is polytopic, the figurate is labeled a polytopic number, and may be a polygonal number or a polyhedral number.  is a number which can be represented by a regular geometrical arrangement of equally spaced points. If the arrangement forms a regular polygon polygon, closed plane figure bounded by straight line segments as sides. A polygon is convex if any two points inside the polygon can be connected by a line segment that does not intersect any side. If a side is intersected, the polygon is called concave.  the number is called a polygonal number In mathematics, a polygonal number is a number that can be arranged as a regular polygon. Ancient mathematicians discovered that numbers could be arranged in certain ways when they were represented by pebbles or seeds; such numbers, which can be made from figures, are generally . Different figurate Fig´ur`ate

a. 1. Of a definite form or figure.
Plants are all figurate and determinate, which inanimate bodies are not.
- Bacon.

2. Figurative; metaphorical.
3. (Mus.
 sequences are formed depending upon the dimension we consider. Each dimension gives rise to a system of figurate sequences which are infinite in number.

In this paper we consider a figurate sequence of Triangular numbers of dimension 3, also called as Pyramidal numbers.

The nth Pyramidal number A pyramidal number is a figurate number that represents a pyramid with a base and a given number of sides. The term is most often used to refer to square pyramidal numbers, which have four sides, but it can also refer to:
  • Triangular pyramidal number (three sides)
 [t.sub.n], n [member of] N is defined by:

[t.sub.n] = n(n+1)(n+2) / 6

We can obtain the first k terms of Pyramidal numbers in Maple as;
> t := n->(1/6)*n*(n+1)*(n+2):
> first :+ k -> seq (t(n), n=1 ... 20):
> first(20);

  1, 4, 10, 20, 35, 56, 84, 120, 165, 220, 286, 364, 455, 560, 680,
        816, 969, 1140, 1330, 1540


Note that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE re·pro·duce  
v. re·pro·duced, re·pro·duc·ing, re·pro·duc·es

v.tr.
1. To produce a counterpart, image, or copy of.

2. Biology To generate (offspring) by sexual or asexual means.
 IN ASCII ASCII or American Standard Code for Information Interchange, a set of codes used to represent letters, numbers, a few symbols, and control characters. Originally designed for teletype operations, it has found wide application in computers. .]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (20)

Combining(17), (19) and (20), we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

with

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

which has a Dirichlet series In mathematics, a Dirichlet series is any series of the form



where s and an, n = 1, 2, 3, ... are complex numbers.
 expansion, absolutely convergent absolutely convergent
adj.
Of, relating to, or characterized by absolute convergence.



absolutely convergent  

Relating to or characterized by absolute convergence.
 for [??]s > 1/5.

By Lemma lemma (lĕm`ə): see theorem.

(logic) lemma - A result already proved, which is needed in the proof of some further result.
 1, for Rs > 1 we have,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (21)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (22)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (23)

For constructing Smarandache sequence of Pyramidal numbers we use the operation of concatenation on the terms of the above sequence.This operation is defined as ;

> conc :=(n,m)-> n*10^length(m)+m:

We define Smarandache consecutive sequence {[scs.sub.n]} for Pyramidal numbers recursively as;
[scs.sub.1] = [u.sub.1],
[scs.sub.n] = conc([scs.sub.n-1], [u.sub.n])


Using MapleWe have obtained first 20 terms of Smarandache consecutive sequence of Pyramidal numbers;
>conc :=(n,m)-> n*10^length(m)+m:
> scs_n := (u,n)-> if n = 1 then u(1)else conc(scs_n(u,n-1),u(n))fi:
> scs := (u,n)-> seq (scs_n(u,i),i=1 ... n):
> scs(t,20);

  1, 14, 1410, 141020, 1402035, 1410203556, 141020355684,
        141020355684120, 141020355684120165, 141020355684120165220,
        141020355684120165220286, 141020355684120165220286364,
        141020355684120165220286364455,
        14102035568412016522028636445560,
        14102035568412016522028636445560680,
        14102035568412016522028636445560680816,
        14102035568412016522028636445560680816969,
        141020355684120165220286364455606808169691140,
        1410203556841201652202863644556068081696911401330,
        14102035568412016522028636445560680816969114013301540,


Display of the same sequence in the triangular form is;
> show := L -> map(i ->print(i),L):
> show([scs(t,20)]);

                        1
                        14
                       1410
                      141020
                     14102035
                    1410203556
                   141020355684
                  141020355684120
                 141020355684120165
               141020355684120165220
              141020355684120165220286
             141020355684120165220286364
            141020355684120165220286364455
          141020355684120165220286364455560
         141020355684120165220286364455560680
       141020355684120165220286364455560680816
      141020355684120165220286364455560680816969
    1410203556841201652202863644555606808169691140
  14102035568412016522028636445556068081696911401330
141020355684120165220286364455560680816969114013301540


The reversed Smarandache sequence (rss)associated with a given sequence {[u.sub.n]}; n [member of] N is defined recursively as:
[rss.sub.1] = [u.sub.1],
[rss.sub.n] = conc([u.sub.n], [rss.sub.n-1]).


In Maple we use the following program;
> rss_n :=(u,n) -> if n=1 then u(1) else conc(u(n),rss_n(u,n-1)) fi:
> rss := (u,n) -> seq(rss_n(u,i),i=1..n):


We get the first 20 terms of reversed smarandache sequence of Pyramidal numbers as;
> rss(t,20);

  1, 41, 1041, 201041, 35201041, 5635201041, 845635201041,
        120845635201041, 165120845635201041, 220165120845635201041,
        286220165120845635201041, 364286220165120845635201041,
        455364286220165120845635201041,
        560455364286220165120845635201041,
        680560455364286220165120845635201041,
        816680560455364286220165120845635201041,
        969816680560455364286220165120845635201041,
        1140969816680560455364286220165120845635201041,
        13301140969816680560455364286220165120845635201041,
        154013301140969816680560455364286220165120845635201041


Smarandache Mirror Sequence (sms)is defined as follows:
[sms.sub.1] = [u.sub.1],
[sms.sub.n] = conc(conc([u.sub.n], [sms.sub.n-1]), [u.sub.n]).


The following program gives first 20 terms of Smarandache Mirror sequence of Pyramidal numbers.
> sms_n := (u,n) -> if n=1 then
> u(1)
> else
> conc(conc(u(n),sms_n(u,n-1)),u(n))
> fi:
> sms :=(u,n) ->seq(sms_n(u,i), i=1..n):
> sms(t,20);

  1, 414, 1041410, 20104141020, 352010414102035, 5635201041410203556,
        84563520104141020355684, 12084563520104141020355684120,
        16512084563520104141020355684120165,
        22016512084563520104141020355684120165220,
        28622016512084563520104141020355684120165220286,
        36428622016512084563520104141020355684120165220286364,
        45536428622016512084563520104141020355684120165220286364455,
        5604553642862201651208456352010414102035568412016522028636        4455560, 68056045536428622016512084563520104141020355684120        165220286364455560680, 816680560455364286220165120845635201        04141020355684120165220286364455560680816. 9698166805604553        6428622016512084563520104141020355684120165220286364455560        680816969, 114096981668056045536428622016512084563520104141        0203556841201652202863644555606808169691140, 13301140969816        6805604553642862201651208456352010414102035568412016522028        636445556068081696911401330, 154013301140969816680560455364        2862201651208456352010414102035568412016522028636445556068        0816969114013301540


Finally Smarandache Symmetric sequence (sss) is defined as:
[sss.sub.2n-1] = conc(bld([scs.sub.2n-1]), [rss.sub.2n-1]),
[sss.sub.2n] = conc([scs.sub.2n], [rss.sub.2n]), n [member of] N,


where the function "bld" (But Last Digit) is defined in Maple as

> bld := n->iquo(n,10):

First 20 terms of Smarandache Symmetric sequence are obtained as
> bld :=n-> iquo(n,10):
> conc := (n,m)-> n*10^length(m)+m:
> sss_n := (u,n) -> if type(n,odd) then
> conc(bld(scs_n(u,(n+1)/2)),rss_n(u,(n+1)/2))
> else
> conc(scs_n(u,n/2),rss_n(u,n/2)
> fi:
> sss := (u,n) -> seq(sss_n(u,i), i=1..n):
> sss(t,20);

  1, 11, 141, 1441, 1411041, 14101041, 14102201041, 141020201041,
        141020335201041, 1410203535201041, 1410203555635201041,
        14102035565635201041, 14102035568845635201041,
        141020355684845635201041, 14102035568412120845635201041,
        141020355684120120845635201041,
        14102035568412016165120845635201041,
        141020355684120165165120845635201041,
        14102035568412016522220165120845635201041,
        141020355684120165220220165120845635201041


We find out primes from a large (first 500) terms of various Smarandache sequences defined so far. We have used Maple 6 on Pentium Pentium

Family of microprocessors developed by Intel Corp. Introduced in 1993 as the successor to Intel's 80486 microprocessor, the Pentium contained two processors on a single chip and about 3.3 million transistors.
 3 with 128Mb RAM. We first collect the lists of first 500 terms of the consecutive, reversed, mirror and symmetric sequences of Pyramidal numbers;
> st :=time(): Lscs500:=[scs(t,500)]: printf("%a seconds",
round(time()-st)); 15 seconds
> st :=time(): Lrss500:=[rss(t,500)]: printf("%a seconds",
round(time()-st)); 20 seconds
> st :=time(): Lsms500:=[sms(t,500)]: printf("%a seconds",
round(time()-st)); 58 seconds
> st :=time(): Lsss500:=[sss(t,500)]: printf("%a seconds",
round(time()-st)); 12 seconds


Further we find the number of digits in the 500th term of each sequence.
> length(Lscs500[500]),length(Lrss500[500]);

3283, 3283

> length(Lsms500[500]),length(Lsss500[500]);

6565, 2846


There exist no prime in the first 500 terms of Smarandache consecutive sequence of Pyramidal numbers;
> st:= time():select(isprime,Lscs500);

[]

> printf("%a minutes",round((time()-st)/60));
9 minutes


There is only one prime in the first 500 terms of reversed Smarandache sequence of Pyramidal numbers;
> st:= time():
> select(isprime,Lrss500);

[41]

> printf("%a minutes",round((time()-st)/60));
119 minutes


There is no prime in the first 500 terms of Smarandache mirror sequence;
> st:= time():
> select(isprime,Lsms500);
> printf("%a minutes",round((time()-st)/60));

[]

177 minutes


There is only one prime in the first 500 terms of Smarandache symmetric sequence;
> st:= time():
> select(isprime,Lsss500);

[11]

> printf("%a minutes",round((time()-st)/60));
90 minutes


[section] 2. Open problems

1) How many Pyramidal numbers are there in the first 500 terms of Smarandache consecutive, mirror, symmetric and reverse symmetric sequences of Pyramidal numbers ?

2) What are those numbers ?

Received Nov. 12, 2006

References

[1] Delfim F.M., Viorica Teca, Consective, Reversed, Mirror and Symmetric Smarandache Sequences of Triangular Numbers, Scientia Magna, 1(2005), No.2, 39-45.

[2] Muktibodh A.S., Smarandache Sequences of Pentagonal Numbers, Scientia Magna, 2(2006), No.3.

[3] Muktibodh A.S., Figurate Systems, Bull. Marathwada Mathematical Soc., 6(2005), No.1, 12-20.

[4] Shyam Sunder Gupta, Smarandache Sequence of Triangular Numbers, Smarandache Notions Journal, 14, 366-368.

Arun Muktibodh, Uzma Sheikh sheikh
 or shaykh

Among Arabic-speaking tribes, especially Bedouin, the male head of the family, as well as of each successively larger social unit making up the tribal structure. The sheikh is generally assisted by an informal tribal council of male elders.
, Dolly Juneja, Rahul Barhate and Yogesh Miglani

Mohota Science College

Umred Rd., Nagpur-440009, India.
COPYRIGHT 2007 American Research Press
No portion of this article can be reproduced without the express written permission from the copyright holder.
Copyright 2007 Gale, Cengage Learning. All rights reserved.

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Author:Muktibodh, Arun; Sheikh, Uzma; Juneja, Dolly; Barhate, Rahul; Miglani, Yogesh
Publication:Scientia Magna
Geographic Code:1USA
Date:Jan 1, 2007
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